In the following problem, the expression is the right side of the formula for [tex]\cos (\alpha-\beta)[/tex] with particular values for [tex]\alpha[/tex] and [tex]\beta[/tex].

[tex]\[ \cos \left(\frac{7 \pi}{12}\right) \cos \left(\frac{5 \pi}{12}\right)+\sin \left(\frac{7 \pi}{12}\right) \sin \left(\frac{5 \pi}{12}\right) \][/tex]

a. Identify [tex]\alpha[/tex] and [tex]\beta[/tex] in the expression.

The value of [tex]\alpha[/tex] is [tex]\square[/tex]

The value of [tex]\beta[/tex] is [tex]\square[/tex]

b. Write the expression as the cosine of an angle.

The expression is [tex]\cos \square[/tex]

c. Find the exact value of the expression.

[tex]\[ \cos \left(\frac{7 \pi}{12}\right) \cos \left(\frac{5 \pi}{12}\right)+\sin \left(\frac{7 \pi}{12}\right) \sin \left(\frac{5 \pi}{12}\right)=\square \][/tex]

(Type an exact answer, using radicals as needed. Use integers or fractions for any numbers in the expression. Rationalize all denominators.)



Answer :

Let's tackle the problem step by step:

### Part a. Identify [tex]$\alpha$[/tex] and [tex]$\beta$[/tex] in the expression.
Given the expression:
[tex]\[ \cos \left(\frac{7 \pi}{12}\right) \cos \left(\frac{5 \pi}{12}\right) + \sin \left(\frac{7 \pi}{12}\right) \sin \left(\frac{5 \pi}{12}\right) \][/tex]
we know this matches the cosine of the difference formula:
[tex]\[ \cos(\alpha - \beta) = \cos(\alpha)\cos(\beta) + \sin(\alpha)\sin(\beta) \][/tex]

By comparing, we identify:
[tex]\[ \alpha = \frac{7\pi}{12} \][/tex]
[tex]\[ \beta = \frac{5\pi}{12} \][/tex]

So:
[tex]\[ \text{The value of } \alpha \text{ is } \frac{7\pi}{12} \][/tex]
[tex]\[ \text{The value of } \beta \text{ is } \frac{5\pi}{12} \][/tex]

### Part b. Write the expression as the cosine of an angle.

The expression can be rewritten using the cosine of the difference formula:
[tex]\[ \cos \left(\frac{7\pi}{12} - \frac{5\pi}{12}\right) \][/tex]

Simplifying the angle inside the cosine function:
[tex]\[ \frac{7\pi}{12} - \frac{5\pi}{12} = \frac{2\pi}{12} = \frac{\pi}{6} \][/tex]

Therefore, the expression simplifies to:
[tex]\[ \cos \left(\frac{\pi}{6}\right) \][/tex]

So:
[tex]\[ \text{The expression is } \cos \left(\frac{\pi}{6}\right) \][/tex]

### Part c. Find the exact value of the expression.

We know from trigonometric identities that:
[tex]\[ \cos \left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2} \][/tex]

Thus, the exact value of the initial expression is:
[tex]\[ \cos \left(\frac{7\pi}{12}\right) \cos \left(\frac{5\pi}{12}\right) + \sin \left(\frac{7\pi}{12}\right) \sin \left(\frac{5\pi}{12}\right) = \frac{\sqrt{3}}{2} \][/tex]

So:
[tex]\[ \cos \left(\frac{7 \pi}{12}\right) \cos \left(\frac{5 \pi}{12}\right) + \sin \left(\frac{7 \pi}{12}\right) \sin \left(\frac{5 \pi}{12}\right) = \frac{\sqrt{3}}{2} \][/tex]